Design and convergence analysis of numerical methods for stochastic evolution equations with Leray-Lions operator

Jérôme Droniou, Beniamin Goldys, Kim Ngan Le

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Abstract

The gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper we study the GDM for a general stochastic evolution problem based on a Leray-Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the gradient scheme (GS) solutions is proved by using discrete functional analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.

Original languageEnglish
Pages (from-to)1143-1179
Number of pages37
JournalIMA Journal of Numerical Analysis
Volume42
Issue number2
DOIs
Publication statusPublished - 1 Apr 2022

Keywords

  • convergence analysis
  • gradient discretization method
  • numerical methods
  • p-Laplace equation
  • stochastic PDE

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